Multiscale Elastic Models of Collagen Bio-structures: From Cross-Linked Molecules to Soft Tissues - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 80

This can be obtained by means of a structural multiscale technique, coupling a
structural rationale with multiscale homogenization approaches. Accordingly, the
equivalent responses of tissue substructures at different scales are analytically
derived and consistently integrated, allowing to include at the macroscale the
dominant mechanisms occurring at smaller scales [ 8 , 50 ].
In this context, the authors recently proposed elastic constitutive models for
collagen-rich tissues, based on multiscale homogenization techniques that
explicitly incorporate nanoscale and microscale mechanisms, as well as their
coupling effects [ 8 , 51 , 52 ]. By simulating histological alterations at nano, micro
and macro scales, present models have been proved to be able to highlight and to
analyze the deep link between histology and mechanical response of both col-
lagenous tissues and body structures [ 52 , 53 ]. Such an approach allows also to
include, through convex analysis arguments and in the same modeling framework,
damage evolution at different scales, induced by both mechanical and non-
mechanical sources [ 15 ].
In the following, a mechanical model of soft collagenous tissues is discussed,
addressing the purely elastic response and neglecting any inelastic and damage
mechanism. The model generalizes the one proposed and applied in [ 8 , 51 ], and
follows a structural multiscale rationale. The macroscale tissue response is
recovered by integrating single-scale models of collagenous bio-structures at very
different length scales: molecules (nanoscale), fibrils (mesoscale) and crimped
fibers (microscale). Following a structural approach, the ordered histology of both
uni-directional and multi-directional tissues is explicitly taken into account.
As a result of a multi-step homogenization procedure, the homogenized tissue
at the macroscale is treated as a non-linearly elastic anisotropic continuum, passing
from its reference configuration to the actual one via a quasi-static deformation
path /, governed by a time-like variable s. In turn, bio-structures at lower scales
undergo to /-induced quasi-static transformations. At each scale this process is
herein described following an incremental strategy. For the sake of notation, in
what follows the symbol x denotes the partial derivative of x with respect to s.

4 Nanoscale Mechanics: Molecules

A collagen molecule is modelled as an equivalent zero-dimensional nano-
structure, whose reference end-to-end length is ' m ; o , and is lower than its contour
length ' c . Let A m be a measure of the molecular cross-sectional area (assumed to
be constant during the overall deformation process), and e m ¼ ' m =' m ; o 1a
measure of the molecular nominal strain, ' m being the actual molecular end-to-end
length.
Entropic and energetic mechanisms are assumed to act as in series and they
contribute to the overall molecular stretch measure e m by e s m

and e m , respectively,

so that by compatibility